- Type about:cache in the Location bar.
- Click "Disk Cache Device", loading this page might take some time.
- Search for the FLV file (Ctrl+F). You may have to look around a bit.
- Once found, click the link, which will open up a "Cache entry information" page.
- On that page, the "file on disk" parameter shows you where Firefox has stored that FLV file in its cache on disk. Now just cp that file to wherever you want it:
cp /path/from/file/on/disk /path/to/where/I/want/FLV/file.flv
Incredibly simple and easy! (Tested on Ubuntu Feisty)